Grassmann Algebra

TheInteriorProduct.nb 42
� The angle between two bivectors
Suppose we have two bivectors x1 � x2 and x3 � x4 .
Diagram of three vectors showing the angles between them, and the angle between the
bivector and the vector.
In a 3-space we can find the vector complements of the bivectors, and then find the angle
between these vectors. This is equivalent to the cross product formulation. Let A be the cosine
of the angle between the vectors, then:
Cos�Θ� � �x1
�������������
� x2�
����
�������������
�x3 � x4�
��������������������������������
������������� ���������������
�x1 �
������������� �
x2���x3
� x4�
�x1 � x2� ���� �x3 � x4�
�������������������������������� ���������������
�x1 � x2���x3 � x4�
But from formulae 6.25 and 6.46 we can remove the complement operations to get the simpler
expression:
Cos�� � �x1 � x2� ���� �x3 � x4�
�������������������������������� ���������������
�x1 � x2���x3 � x4�
6.115
Note that, in contradistinction to the 3-space formulation using cross products, this formulation
is valid in a space of any number of dimensions.
An actual calculation is most readably expressed by expanding each of the terms separately. We
can either represent the xi in terms of basis elements or deal with them directly. A direct
expansion, valid for any metric is:
ToScalarProducts��x1 � x2� ���� �x3 � x4��
��x1 � x4� �x2 � x3� � �x1 � x3� �x2 � x4�
Measure�x1 � x2� Measure�x3 � x4�
���������������������������������������������������������������������������
��x1 � x2� 2 � �x1 � x1� �x2 � x2� ���������������������������������������������������������������������������
��x3 � x4� 2 � �x3 � x3� �x4 � x4�
� The volume of a parallelepiped
We can calculate the volume of a parallelepiped as the measure of the trivector whose vectors
make up the sides of the parallelepiped. GrassmannAlgebra provides the function Measure for
expressing the volume in terms of scalar products:
2001 4 5
V � Measure�Α1 � Α2 � Α3�
�
���Α1 � Α3�2 �Α2 � Α2� �
2 �Α1 � Α2� �Α1 � Α3� �Α2 � Α3� � �Α1 � Α1� �Α2 � Α3� 2 �
�Α1 � Α2�2 �Α3 � Α3� � �Α1 � Α1� �Α2 � Α2� �Α3 � Α3��